#1
I have a pretty good understanding of basic maths (highschool) and I was actually good with more advanced maths like integrals (university) a while ago, something like 10 years ago. The thing is I'm really rusted and I'd like to get back at it, but I don't know where to start. I feel lost. I would love to go in deep with pure mathematics. Where should I start?
Thanxs
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#2
You could do my homework for me.
Quote by DisarmGoliath
Facesitting is a violation of freedom of speech, because how can you speak when you have an ass covering your face?
#4
Khan's Academy is a free resource that works well if you're in need of something to supplement your studying. I don't know how well it would work alone, but I'm sure it could be helpful
#6
Video lectures, such as the ones from the MIT.
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#7
Quote by /-Vince-\
I would love to go in deep with pure mathematics.

Get yourself a calculus textbook and go in deep and raw.
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#11
the purest form of mathematics is in the measuring (that is, the size) of one's penis
#12
Quote by TwinkyFist
I don't even actually know what pure mathematics is.


mathematics presented without concern for its applications. this also includes math without applications which is the good kind hmu
i don't know why i feel so dry
#14
application ≠ rules
it's all just coming back
it's all coming back

it's all coming back to me
#15
Quote by megano28
Khan's Academy is a free resource that works well if you're in need of something to supplement your studying. I don't know how well it would work alone, but I'm sure it could be helpful

Khan's isn't the greatest at explaining things well.
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#16
inb4 eastwinn

edit: when did you stop being through being cool? now i look like a goof good one pal
Last edited by MinterMan22 at Dec 5, 2014,
#17
Quote by MinterMan22
inb4 eastwinn

**** you
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#18
Quote by TwinkyFist
So, math with no rules. 2+2 can equal fish. I see it now.


sort of? i mean i know this is a joke, but i think it's a good point. mathematics is built on axioms, or so has been the tradition for nearly 2000 years. axioms are the rules everyone agrees on to start off with. if you took geometry in high school (or later) you probably learned euclid's axioms which, as far as we can tell, are supposed to represent the real world. the development of the set theory in the late 1800s gives us the axioms that modern mathematicians take as the rules with limited controversy. if you're feeling ambitious, you can read them here.

that said, there's nothing stopping you from changing the rules. well, you should mention that you are otherwise everyone will think you're an idiot. from these new rules you can deduce what is now true that wasn't before and what's false now that wasn't before.

to give a historical example, take euclid's axioms

1. "To draw a straight line from any point to any point."
2. "To produce [extend] a finite straight line continuously in a straight line."
3. "To describe a circle with any centre and distance [radius]."
4. "That all right angles are equal to one another."
5. "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."

the fifth is frequent restated, "given a line and a point not on that line, one and only one line may be drawn through that point such that it does not intersect with the original line." euclid also gave his "common notions" which are really just more axioms, all things considered. they are


1. Things that are equal to the same thing are also equal to one another (Transitive property of equality).
2. If equals are added to equals, then the wholes are equal (Addition property of equality).
3. If equals are subtracted from equals, then the remainders are equal (Subtraction property of equality).
4. Things that coincide with one another are equal to one another (Reflexive Property).
5. The whole is greater than the part.

(i'm just copying these from wiki)

euclid also provides some standard definitions. he then uses these axioms, notions (really just axioms but he gave them a different name), and definitions hto construct geometry as most of you all know it. the idea here is that the rules exist, and he found them, rather than made them up on the spot.

there are some issues though. for one thing, these axioms are not sufficient. you can, for example, prove that all triangles are equilateral if you're sneaky. later mathematicians added to the rules and defined them so this can't happen. at the same time, they wanted to do away with that 5th axiom, the long one. they figured it could be proven using the other rules. with some big ideas i can't begin to describe here, it was proven that given the other rules, that long fifth one cannot be proven or disproven. it must be taken as a rule.

so some cheeky mathematicians, in particular a dude named guass, decided to see what would happen if you just didn't use it. euclid originally uses his 5th axiom to show that all the inside angles of a triangle add up to 180 degrees. if you deny the 5th axiom, it must be possible to construct a triangle whose angles add up to something other than 180 degrees.

this is the case on a sphere. if you grab a globe, a sharpie, and a nice right angle, you can draw on the globe a triangle with 2 right angles. indeed, drawing on the surface of 3d shapes will break the 5th axiom (cf. topology). but just fiddling with the rules can give you geometries that don't have analogies like this. studying them was merely out of interest. there were no applications.

nowadays there is an application of a non-euclidian geometry (that's what we call ones that don't use the 5th axiom) that also has no analogy on a shape. it came when this dude named lorentz came up with a transformation (a method for turning shapes into other shapes) for understanding objects moving relative to one another. it modeled the universe the way most people did back then as 3 spacial dimensions (with the 5th axiom) that varied over time. this was some pretty rock solid shit but it couldn't account for how people measure the speed of light to be the same no matter how fast they are moving. some fuckboi named einstein thought instead to consider the 3 spacial dimensions and 1 time component as 4 dimensions of a non-euclidean space. in other words, he threw the 5th axiom out. this solved the issue and thus the theory of special relatively was born.

so what i'm saying is that yes, sure, make up new rules. these people did it and it worked out for them. but changing the rules doesn't always lead to anything worthwhile. i mean 2+2=fish? nah, let me show you a rule change that is worth while. here's a new addition table:

+ | 0 | 1 | 2
0 | 0 | 1 | 2
1 | 1 | 2 | 0
2 | 2 | 0 | 1


here 2 + 2 = 1. to see why it's worth while, remember how to do long division with remainders. for example, 101 divided by 3 is 33 remainder 2. likewise, 207 divided by 3 is 69 remainder 2. what's (101 + 207) divided by 3? what happened to the remainder?
i don't know why i feel so dry
#19
now thats what i call mathematics volume 37
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as i walk through the chalet of the shadow of death
everything that you've come to expect


#20
I think my suggestion is the best by far.
Quote by DisarmGoliath
Facesitting is a violation of freedom of speech, because how can you speak when you have an ass covering your face?
#21
Quote by Eastwinn
sort of? i mean i know this is a joke, but i think it's a good point. mathematics is built on axioms, or so has been the tradition for nearly 2000 years. axioms are the rules everyone agrees on to start off with. if you took geometry in high school (or later) you probably learned euclid's axioms which, as far as we can tell, are supposed to represent the real world. the development of the set theory in the late 1800s gives us the axioms that modern mathematicians take as the rules with limited controversy. if you're feeling ambitious, you can read them here...


This is so far over my head. I am so bad at math
#22
Quote by Eastwinn
sort of? i mean i know this is a joke, but i think it's a good point. mathematics is built on axioms, or so has been the tradition for nearly 2000 years. axioms are the rules everyone agrees on to start off with. if you took geometry in high school (or later) you probably learned euclid's axioms which, as far as we can tell, are supposed to represent the real world. the development of the set theory in the late 1800s gives us the axioms that modern mathematicians take as the rules with limited controversy. if you're feeling ambitious, you can read them here.

that said, there's nothing stopping you from changing the rules. well, you should mention that you are otherwise everyone will think you're an idiot. from these new rules you can deduce what is now true that wasn't before and what's false now that wasn't before.

to give a historical example, take euclid's axioms

1. "To draw a straight line from any point to any point."
2. "To produce [extend] a finite straight line continuously in a straight line."
3. "To describe a circle with any centre and distance [radius]."
4. "That all right angles are equal to one another."
5. "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."

the fifth is frequent restated, "given a line and a point not on that line, one and only one line may be drawn through that point such that it does not intersect with the original line." euclid also gave his "common notions" which are really just more axioms, all things considered. they are


1. Things that are equal to the same thing are also equal to one another (Transitive property of equality).
2. If equals are added to equals, then the wholes are equal (Addition property of equality).
3. If equals are subtracted from equals, then the remainders are equal (Subtraction property of equality).
4. Things that coincide with one another are equal to one another (Reflexive Property).
5. The whole is greater than the part.

(i'm just copying these from wiki)

euclid also provides some standard definitions. he then uses these axioms, notions (really just axioms but he gave them a different name), and definitions hto construct geometry as most of you all know it. the idea here is that the rules exist, and he found them, rather than made them up on the spot.

there are some issues though. for one thing, these axioms are not sufficient. you can, for example, prove that all triangles are equilateral if you're sneaky. later mathematicians added to the rules and defined them so this can't happen. at the same time, they wanted to do away with that 5th axiom, the long one. they figured it could be proven using the other rules. with some big ideas i can't begin to describe here, it was proven that given the other rules, that long fifth one cannot be proven or disproven. it must be taken as a rule.

so some cheeky mathematicians, in particular a dude named guass, decided to see what would happen if you just didn't use it. euclid originally uses his 5th axiom to show that all the inside angles of a triangle add up to 180 degrees. if you deny the 5th axiom, it must be possible to construct a triangle whose angles add up to something other than 180 degrees.

this is the case on a sphere. if you grab a globe, a sharpie, and a nice right angle, you can draw on the globe a triangle with 2 right angles. indeed, drawing on the surface of 3d shapes will break the 5th axiom (cf. topology). but just fiddling with the rules can give you geometries that don't have analogies like this. studying them was merely out of interest. there were no applications.

nowadays there is an application of a non-euclidian geometry (that's what we call ones that don't use the 5th axiom) that also has no analogy on a shape. it came when this dude named lorentz came up with a transformation (a method for turning shapes into other shapes) for understanding objects moving relative to one another. it modeled the universe the way most people did back then as 3 spacial dimensions (with the 5th axiom) that varied over time. this was some pretty rock solid shit but it couldn't account for how people measure the speed of light to be the same no matter how fast they are moving. some fuckboi named einstein thought instead to consider the 3 spacial dimensions and 1 time component as 4 dimensions of a non-euclidean space. in other words, he threw the 5th axiom out. this solved the issue and thus the theory of special relatively was born.

so what i'm saying is that yes, sure, make up new rules. these people did it and it worked out for them. but changing the rules doesn't always lead to anything worthwhile. i mean 2+2=fish? nah, let me show you a rule change that is worth while. here's a new addition table:

+ | 0 | 1 | 2
0 | 0 | 1 | 2
1 | 1 | 2 | 0
2 | 2 | 0 | 1


here 2 + 2 = 1. to see why it's worth while, remember how to do long division with remainders. for example, 101 divided by 3 is 33 remainder 2. likewise, 207 divided by 3 is 69 remainder 2. what's (101 + 207) divided by 3? what happened to the remainder?

I could've said all this but in fewer words, so I think I win?
it's all just coming back
it's all coming back

it's all coming back to me
#23
The mathematics that I had learned in my yonder years were impure practices of mathematicianry. Y axis's were replaced with Z axis's and Z axis's were replaced with y axis's to recreate the logical fallacy that no one will ever understand math, even in its purest form, the pure mathematics that you speak of.
Most of the important things


in the world have been accomplished


by people who have kept on


trying when there seemed to be no hope at all
#24
Quote by TwinkyFist
This is so far over my head. I am so bad at math


those axioms are one of things that everyone pretends to understand but few people actually do

i will go on pretending though
i don't know why i feel so dry
#25
I saw somewhere some mentioned learn matlab, I second this as an engineer + applied maths student matlab is fantastic. Not only does it provide a useful way to verify your answer and compute numerical methods, but it also seriously reinforces understanding concepts because you're forced to code it. This is as opposed to just learning symbolic manipulations, which is no good or fun as you get deeper in to maths. I'd suggest reading good academic literature, not the stupid Stewart calculus book that everywhere seems to use, that really has some good exposition and explanation of the why of the maths. Texts that I own that I find to be really useful Calculus by Michael Spivak (has loads of good juicy meat that is helping me get into real analysis), div grad curl and all that by schey ( a fantastic guide into vector analysis).

But in terms of where to start in skill, start from the bottom. Look at basic algebra, check and see if you understand and can apply it well, then do that with just about every subject after, calculus follows itself quite well into its numerous subdivisions. I do highly suggest learning linear algebra either before reintroducing yourself to calculus, as it does better job of explaining certain things about calculus much better than calculus can itself.
#26
I used MatLab once to simulate the EPR spectrum of some simple conjugated molecule. It was very confusing.

Buy an A-Level maths textbook or something and work your way through that is what I would try to do.

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